Note on the quadratic Gauss sums (Q5932786)

Let \(p > 2\) be a prime number and \( \chi(x)=(x/p)\) the Legendre symbol. The author shows that the quadratic Gauss sums NEWLINE\[NEWLINEG(k,\chi)=\sum_{x=0}^{p-1}\chi(x) e^{2 \pi i x/p}, \qquad k=0,1 , \ldots , p-1,NEWLINE\]NEWLINE are the eigenvalues of the circulant \(p\times p\) matrix NEWLINE\[NEWLINE X=\left( \begin{matrix} \chi(0) & \chi(1) & \ldots & \chi(p-1)\cr \chi(p-1)&\chi(0)&\ldots&\chi(p-2)\cr \vdots&\vdots&\ddots&\vdots\cr \chi(1)&\chi(2)&\ldots&\chi(0)\cr \end{matrix}\right) . NEWLINE\]NEWLINE It should be noted that the author's argument is not new. A similar representation of \(G(k,\chi)\) was used by I. Schur in his well-known proof of the Gauss theorem concerning the exact value of \(G(1,\chi)\): NEWLINE\[NEWLINEG(1,\chi)= \begin{cases} \sqrt{p},& \text{if }p\equiv 1\pmod 4;\\ i\sqrt{p},& \text{if }p\equiv 3\pmod 4\end{cases}NEWLINE\]NEWLINE (for example, see Exercises 24-25 on pages 19-20 in the reviewer's book: ``Arithmetic of Algebraic Curves'', Plenum, New York (1994; Zbl 0862.11036)).